1 00:00:00,000 --> 00:00:03,780 The following content is provided by MIT OpenCourseWare 2 00:00:03,780 --> 00:00:05,110 under a Creative Commons license. 3 00:00:05,110 --> 00:00:08,585 Additional information about our license, and MIT 4 00:00:08,585 --> 00:00:12,541 OpenCourseWare in general is available at ocw.mit.edu. 5 00:00:12,541 --> 00:00:15,340 6 00:00:15,340 --> 00:00:16,530 PROFESSOR: Vasily. 7 00:00:16,530 --> 00:00:22,700 Vasily Strela who works now for Morgan Stanley, did his 8 00:00:22,700 --> 00:00:27,000 PhD here in the math department, and kindly said he 9 00:00:27,000 --> 00:00:30,610 would tell us about financial mathematics. 10 00:00:30,610 --> 00:00:32,730 So, it's all yours. 11 00:00:32,730 --> 00:00:36,450 12 00:00:36,450 --> 00:00:39,690 GUEST SPEAKER: Let me thank Professor Strang for giving 13 00:00:39,690 --> 00:00:43,380 this opportunity to talk here, and it feels very good to be 14 00:00:43,380 --> 00:00:46,930 back, be back to 18.086. 15 00:00:46,930 --> 00:00:50,420 So, a few more words about myself. 16 00:00:50,420 --> 00:00:54,440 I've been Professor Strang's student in mathematics about 17 00:00:54,440 --> 00:00:56,060 ten years ago. 18 00:00:56,060 --> 00:00:59,790 So after receiving my PhD, I taught 19 00:00:59,790 --> 00:01:02,870 mathematics for a few years. 20 00:01:02,870 --> 00:01:07,590 Then ended up working for a financial institution for 21 00:01:07,590 --> 00:01:11,660 investment bank, Morgan Stanley in particular. 22 00:01:11,660 --> 00:01:15,600 I'm part of an analytic modeling group in 23 00:01:15,600 --> 00:01:17,950 fixed income division. 24 00:01:17,950 --> 00:01:22,950 What we are doing, we are doing multiplications in 25 00:01:22,950 --> 00:01:26,910 finance and modeling derivatives, fixed income 26 00:01:26,910 --> 00:01:28,970 derivatives. 27 00:01:28,970 --> 00:01:31,530 That's actually what I'm going to talk about today. 28 00:01:31,530 --> 00:01:36,690 I want to show how 18.086, it's a wonderful class which I 29 00:01:36,690 --> 00:01:41,740 admire a lot, which applications it has in the 30 00:01:41,740 --> 00:01:44,720 real world, and in particular in finance 31 00:01:44,720 --> 00:01:47,650 and derivatives pricing. 32 00:01:47,650 --> 00:01:51,720 Let's start with a simple example, which actually comes 33 00:01:51,720 --> 00:01:57,050 not from finance, but rather from gambling. 34 00:01:57,050 --> 00:02:03,580 Well, let's look at horse racing or cockroach racing, if 35 00:02:03,580 --> 00:02:06,180 you prefer. 36 00:02:06,180 --> 00:02:10,200 Suppose there are two horses, and sure enough people bet on 37 00:02:10,200 --> 00:02:15,270 them and bookie uses a clever one, very scientific-minded 38 00:02:15,270 --> 00:02:20,120 guy and he made a very good research of previous history 39 00:02:20,120 --> 00:02:22,840 of these two horses. 40 00:02:22,840 --> 00:02:28,980 He found out that the first horse has 20% chance to win. 41 00:02:28,980 --> 00:02:32,660 The second horse has 80% chance to win. 42 00:02:32,660 --> 00:02:36,100 He is actually right about his knowledge 43 00:02:36,100 --> 00:02:38,250 about chances to win. 44 00:02:38,250 --> 00:02:41,840 When that becomes general public, people who bet, they 45 00:02:41,840 --> 00:02:45,680 don't have access to all information, and the bets are 46 00:02:45,680 --> 00:02:46,870 split slightly differently. 47 00:02:46,870 --> 00:02:54,590 So the 10,000 is placed on the first horse, and 50,000 is 48 00:02:54,590 --> 00:02:57,970 placed on the second horse. 49 00:02:57,970 --> 00:03:02,040 Bookie, sticking to his scientific knowledge, places 50 00:03:02,040 --> 00:03:03,900 the odds 4:1. 51 00:03:03,900 --> 00:03:07,990 Meaning that if the first horse wins then whoever put on 52 00:03:07,990 --> 00:03:12,130 the horse gets his money back and four times his money back 53 00:03:12,130 --> 00:03:13,370 on top of it. 54 00:03:13,370 --> 00:03:16,910 Or if the second horse wins then whoever put money on this 55 00:03:16,910 --> 00:03:24,190 horse will get the money back and 1/4 on top of it. 56 00:03:24,190 --> 00:03:26,360 So, let's see. 57 00:03:26,360 --> 00:03:34,460 What are chances for bookie to win or lose in this situation? 58 00:03:34,460 --> 00:03:38,760 Well, if the first horse wins then he has to give back 59 00:03:38,760 --> 00:03:42,890 10,000 plus 40,000, 50,000, and he got 60 00:03:42,890 --> 00:03:46,500 60,000, so he gains 10,000. 61 00:03:46,500 --> 00:03:48,450 Well, good, good for him. 62 00:03:48,450 --> 00:03:52,140 On the other hand, if the second horse wins, then he has 63 00:03:52,140 --> 00:04:00,570 to give back 50,000 plus 1/4 of 50, which is 12,500, so 64 00:04:00,570 --> 00:04:10,210 62.50 altogether, and he loses $2,500. 65 00:04:10,210 --> 00:04:15,320 After many runs, the expected win or loss of the bookmaker 66 00:04:15,320 --> 00:04:18,270 is the probability of the first horse to win times the 67 00:04:18,270 --> 00:04:21,276 expected win, plus the probability of second horse to 68 00:04:21,276 --> 00:04:23,830 win times the expected loss, which turns out 69 00:04:23,830 --> 00:04:25,770 to be exactly zero. 70 00:04:25,770 --> 00:04:35,160 So in each particular run, bookie may lose or win, but in 71 00:04:35,160 --> 00:04:38,390 the long run he expects to break even. 72 00:04:38,390 --> 00:04:42,510 On the other hand, if he would put the chances, he would set 73 00:04:42,510 --> 00:04:47,690 the odds according to the money bet, 5:1, what would be 74 00:04:47,690 --> 00:04:48,790 the outcome? 75 00:04:48,790 --> 00:04:53,690 Well, if the first horse wins he gives back 10,000 plus 76 00:04:53,690 --> 00:05:00,820 50,000, 60,000 exactly the amount he collected. 77 00:05:00,820 --> 00:05:06,290 Or if the second horse wins, again, he gives back 50 plus 78 00:05:06,290 --> 00:05:09,390 1/5 of that 60, he breaks even. 79 00:05:09,390 --> 00:05:14,450 So no matter which horse wins in this scenario, the bookie 80 00:05:14,450 --> 00:05:15,890 breaks even. 81 00:05:15,890 --> 00:05:19,630 How bookie operates, well he actually charges a fee for 82 00:05:19,630 --> 00:05:21,070 each bet, right. 83 00:05:21,070 --> 00:05:25,200 So the second situation is much more preferable for him. 84 00:05:25,200 --> 00:05:28,750 When he doesn't care which horse wins, he just 85 00:05:28,750 --> 00:05:30,520 collects the fee. 86 00:05:30,520 --> 00:05:33,830 Well here, he may lose or gain money. 87 00:05:33,830 --> 00:05:38,870 This is quite beautiful observation, which we will see 88 00:05:38,870 --> 00:05:43,360 how it works in derivatives. 89 00:05:43,360 --> 00:05:47,240 So now back to finance, back to derivatives. 90 00:05:47,240 --> 00:05:50,750 So we are actually interested in pricing a few financial 91 00:05:50,750 --> 00:05:52,880 derivatives, and what is a financial derivative? 92 00:05:52,880 --> 00:05:57,540 Well, a financial derivative is a contract, pay of which at 93 00:05:57,540 --> 00:06:03,530 maturity at some time c, depends on underlying security 94 00:06:03,530 --> 00:06:07,110 -- in our case, we always we will be talking about a stock 95 00:06:07,110 --> 00:06:13,250 as underlying security, and probably interest rates. 96 00:06:13,250 --> 00:06:17,500 What are the examples of financial derivatives? 97 00:06:17,500 --> 00:06:20,590 Well, the most simple example is where it's probably a 98 00:06:20,590 --> 00:06:22,090 forward contract. 99 00:06:22,090 --> 00:06:26,010 Forward contracts is a contract when you agree to 100 00:06:26,010 --> 00:06:30,960 purchase the security for a price which is set today -- 101 00:06:30,960 --> 00:06:34,100 you've agreed to purchase the security in the future for the 102 00:06:34,100 --> 00:06:35,890 price agreed today. 103 00:06:35,890 --> 00:06:41,610 Well, for example, if you needed 1,000 barrel of oil to 104 00:06:41,610 --> 00:06:45,850 keep your house, but not today but rather for the next 105 00:06:45,850 --> 00:06:50,120 winter, on that account you don't want to take the risks 106 00:06:50,120 --> 00:06:55,300 of waiting until the next winter and buying oil then, 107 00:06:55,300 --> 00:07:01,330 you would rather agree on the price now and pay it in the 108 00:07:01,330 --> 00:07:03,880 future and get the oil. 109 00:07:03,880 --> 00:07:05,300 What the price should be? 110 00:07:05,300 --> 00:07:10,590 What is the fair price for this contract? 111 00:07:10,590 --> 00:07:12,500 Well, we will see how to price it. 112 00:07:12,500 --> 00:07:17,820 Well, the few observation here is that this line represents 113 00:07:17,820 --> 00:07:19,190 the payout -- 114 00:07:19,190 --> 00:07:21,440 it's always useful to represent the payout 115 00:07:21,440 --> 00:07:22,700 graphically. 116 00:07:22,700 --> 00:07:24,815 This is just a straight line because the payout of our 117 00:07:24,815 --> 00:07:27,720 contract is f minus k at time c. 118 00:07:27,720 --> 00:07:30,950 This actually gives the current price of the contract 119 00:07:30,950 --> 00:07:35,570 for all different values of the underlying. 120 00:07:35,570 --> 00:07:39,850 Usually, the price of the forward contract is set such 121 00:07:39,850 --> 00:07:43,350 that for the current value of the underlying, the price of 122 00:07:43,350 --> 00:07:45,010 the contract is zero. 123 00:07:45,010 --> 00:07:48,350 It costs nothing to enter a forward contract, so that's 124 00:07:48,350 --> 00:07:52,470 why it intersects zero here. 125 00:07:52,470 --> 00:07:56,690 What are other common derivatives? 126 00:07:56,690 --> 00:08:01,400 Another common derivative are call, and I put European call 127 00:08:01,400 --> 00:08:03,210 input here. 128 00:08:03,210 --> 00:08:05,300 Don't be confused by European or American. 129 00:08:05,300 --> 00:08:08,500 It has nothing to do with Europe or America, it has to 130 00:08:08,500 --> 00:08:12,450 do with the structure of the contract. 131 00:08:12,450 --> 00:08:17,120 European basically means that the contract expires at 132 00:08:17,120 --> 00:08:18,500 certain time c. 133 00:08:18,500 --> 00:08:21,620 American means that's you can exercise this contract at any 134 00:08:21,620 --> 00:08:24,690 time between now and future. 135 00:08:24,690 --> 00:08:27,270 We'll be talking only about European contracts. 136 00:08:27,270 --> 00:08:33,580 So European call option is the contract which gives you the 137 00:08:33,580 --> 00:08:37,130 right, but not obligation, to purchase the underlying 138 00:08:37,130 --> 00:08:42,450 security at set price, k, which is called strike price, 139 00:08:42,450 --> 00:08:46,020 at a future, time c, which is expiration time. 140 00:08:46,020 --> 00:08:52,710 So if your security at time c ends up below k, below the 141 00:08:52,710 --> 00:08:58,860 strike, then sure enough there is no point of buying the 142 00:08:58,860 --> 00:09:01,390 security for a more expensive price. 143 00:09:01,390 --> 00:09:04,170 So the contract expires worthless. 144 00:09:04,170 --> 00:09:08,440 On the other hand, if your stock ends up being greater 145 00:09:08,440 --> 00:09:14,040 than k at expiration time c, then you would make money but 146 00:09:14,040 --> 00:09:18,480 my purchasing this stock for k dollars and your payout will 147 00:09:18,480 --> 00:09:22,100 be f minus k , and this is a graph of your payout. 148 00:09:22,100 --> 00:09:28,180 This line here, as we will see, is the current price of 149 00:09:28,180 --> 00:09:31,670 the contract, and we'll see how to obtain this line in a 150 00:09:31,670 --> 00:09:33,880 few minutes. 151 00:09:33,880 --> 00:09:38,250 Another common contract is a put. 152 00:09:38,250 --> 00:09:43,750 While call was basically a bet that your stock will grow, 153 00:09:43,750 --> 00:09:48,640 right, the put is the bet that your stock will not grow. 154 00:09:48,640 --> 00:09:53,440 So, in this case, the put is the right, but not the 155 00:09:53,440 --> 00:09:58,260 obligation to sell the stock for a certain price k. 156 00:09:58,260 --> 00:10:03,440 Here is the payout, which is similar to the 157 00:10:03,440 --> 00:10:06,830 put, but just flipped. 158 00:10:06,830 --> 00:10:10,250 This is the current price of a put option. 159 00:10:10,250 --> 00:10:16,520 Calls and puts, being very common contracts are traded on 160 00:10:16,520 --> 00:10:17,620 exchanges -- 161 00:10:17,620 --> 00:10:23,790 Chicago exchange is probably the most common place for the 162 00:10:23,790 --> 00:10:27,040 calls and puts on stocks to trade. 163 00:10:27,040 --> 00:10:31,690 I just printed out a Bloomberg screen, which gives the 164 00:10:31,690 --> 00:10:36,370 information about a calls and puts on IBM stock. 165 00:10:36,370 --> 00:10:42,100 So I did it on March 8, and the IBM stock was trading at 166 00:10:42,100 --> 00:10:46,370 this time at $81.14, and here are descriptions of the 167 00:10:46,370 --> 00:10:50,600 contract, they expire on 22nd of April, so it's pretty 168 00:10:50,600 --> 00:10:52,520 short-dated contract. 169 00:10:52,520 --> 00:10:56,420 They can go as far as two years from now, usually. 170 00:10:56,420 --> 00:11:01,990 Here is a set of strikes, and here are a set of prices. 171 00:11:01,990 --> 00:11:05,760 As you can see, there is no single price, there is always 172 00:11:05,760 --> 00:11:08,730 a bid and ask, and that's how dealers and brokers make their 173 00:11:08,730 --> 00:11:13,520 money -- like a bookie, they basically charge you a fee for 174 00:11:13,520 --> 00:11:16,390 selling or buying the contract. 175 00:11:16,390 --> 00:11:21,420 176 00:11:21,420 --> 00:11:24,280 That's how the money are made -- they are made on this 177 00:11:24,280 --> 00:11:27,390 spread, but not on the price of the contract itself, 178 00:11:27,390 --> 00:11:31,060 because as we will see in a second, we actually can price 179 00:11:31,060 --> 00:11:35,150 the contract exactly, and there is no uncertainty once 180 00:11:35,150 --> 00:11:37,340 the price of the stock is set. 181 00:11:37,340 --> 00:11:41,680 182 00:11:41,680 --> 00:11:44,460 There are plenty of other options. 183 00:11:44,460 --> 00:11:47,230 Slightly more exotics contracts, either digital 184 00:11:47,230 --> 00:11:50,470 which pays either zero or one depending on where 185 00:11:50,470 --> 00:11:52,770 your stock ends up. 186 00:11:52,770 --> 00:11:55,160 It probably is not exchange-rated, also -- 187 00:11:55,160 --> 00:11:56,840 I'm not sure. 188 00:11:56,840 --> 00:12:02,110 There are hundreds, if not thousands, of exotic options 189 00:12:02,110 --> 00:12:06,520 where you can say that well, how much would be the right to 190 00:12:06,520 --> 00:12:10,440 purchase a stock for the maximum price between today 191 00:12:10,440 --> 00:12:13,140 and two years from now. 192 00:12:13,140 --> 00:12:16,060 So it will be past-dependent, depending on how the stock 193 00:12:16,060 --> 00:12:20,300 will go, the payout will be defined by this path. 194 00:12:20,300 --> 00:12:23,685 There are American options where you can decide your 195 00:12:23,685 --> 00:12:26,320 option any time between now and maturity, and 196 00:12:26,320 --> 00:12:27,110 so on and so forth. 197 00:12:27,110 --> 00:12:35,700 So, just before we go into mathematics of pricing, just a 198 00:12:35,700 --> 00:12:40,600 few observations and statements. 199 00:12:40,600 --> 00:12:44,470 First of all, it turns out that thanks to developed 200 00:12:44,470 --> 00:12:49,190 mathematics, mathematical theory, if you make certain 201 00:12:49,190 --> 00:12:53,430 assumptions on the dynamics of the stock, then there is no 202 00:12:53,430 --> 00:12:56,390 uncertainty in the price of the option. 203 00:12:56,390 --> 00:13:02,010 You can say exactly how much the option costs now, and 204 00:13:02,010 --> 00:13:06,760 that's what provides, and this is a big 205 00:13:06,760 --> 00:13:08,020 driver for the market. 206 00:13:08,020 --> 00:13:13,270 So dealers quote these contracts and there is a great 207 00:13:13,270 --> 00:13:17,380 agreement on the prices. 208 00:13:17,380 --> 00:13:24,410 The price of the derivative contract is defined completely 209 00:13:24,410 --> 00:13:28,030 by the stock price and not by risk preferences of the market 210 00:13:28,030 --> 00:13:29,820 participant. 211 00:13:29,820 --> 00:13:35,060 So it doesn't matter what are your views on the growth 212 00:13:35,060 --> 00:13:38,080 prospects of the stock. 213 00:13:38,080 --> 00:13:44,150 It will not effect the price of the derivative contract. 214 00:13:44,150 --> 00:13:49,420 As I said, so the mathematical part of it comes into giving 215 00:13:49,420 --> 00:13:52,190 the exact price without any uncertainty. 216 00:13:52,190 --> 00:13:55,080 217 00:13:55,080 --> 00:13:59,880 So let's consider a simple example now. 218 00:13:59,880 --> 00:14:02,690 Let's assume that we are in a very simple world. 219 00:14:02,690 --> 00:14:06,350 Well, first of all, in our world there are only three 220 00:14:06,350 --> 00:14:10,560 options -- the stock itself, the riskless with money market 221 00:14:10,560 --> 00:14:15,550 account, meaning that it is an account where we can either 222 00:14:15,550 --> 00:14:21,300 borrow money or invest money at the request rate r, and 223 00:14:21,300 --> 00:14:22,800 finally our derivative contract. 224 00:14:22,800 --> 00:14:25,720 Here we are not making any assumptions of what kind of 225 00:14:25,720 --> 00:14:27,290 derivative contract it is -- 226 00:14:27,290 --> 00:14:31,670 it could be forward, it could be call, it could be put, it 227 00:14:31,670 --> 00:14:33,280 can be anything. 228 00:14:33,280 --> 00:14:35,980 Moreover, our world is so simple, but first of all, it's 229 00:14:35,980 --> 00:14:41,540 discrete, and second of all, there is only one time step to 230 00:14:41,540 --> 00:14:44,360 the expiration of power of contract, dt. 231 00:14:44,360 --> 00:14:48,990 Not only there is only one step left, we actually know 232 00:14:48,990 --> 00:14:51,690 exactly what our transition probabilities. 233 00:14:51,690 --> 00:14:54,340 There are only two states at the end, and we know the 234 00:14:54,340 --> 00:14:55,170 transition probability. 235 00:14:55,170 --> 00:14:58,650 So with probability p, we move from the state zero to the 236 00:14:58,650 --> 00:15:02,030 state one, and with probability of one minus p, we 237 00:15:02,030 --> 00:15:04,750 move to the state two. 238 00:15:04,750 --> 00:15:07,710 And I just noticed, because this is riskless money market 239 00:15:07,710 --> 00:15:10,670 account, it's the same in both cases. 240 00:15:10,670 --> 00:15:15,580 You just invest money and it grows with 241 00:15:15,580 --> 00:15:18,420 risk-free interest rate. 242 00:15:18,420 --> 00:15:23,830 So, what can I say it about the price of our derivative f? 243 00:15:23,830 --> 00:15:25,300 Well it's simple-minded -- 244 00:15:25,300 --> 00:15:29,310 well, let's start with the forward contract. 245 00:15:29,310 --> 00:15:32,080 We know what the payout in delta t of our forward 246 00:15:32,080 --> 00:15:34,810 contract will be, it will be just the difference between 247 00:15:34,810 --> 00:15:39,390 the stock price and our strike. 248 00:15:39,390 --> 00:15:43,680 Well, a simple-minded approach would be -- well, we know the 249 00:15:43,680 --> 00:15:46,960 transition probabilities, let's just compute the 250 00:15:46,960 --> 00:15:49,740 expected value of our contract, and that's what we 251 00:15:49,740 --> 00:15:55,820 would expect to get if there were many such experiments. 252 00:15:55,820 --> 00:15:58,630 Well, you take the probability of going to state one, you 253 00:15:58,630 --> 00:16:02,230 multiply by the payoff at stage one, take minus p for 254 00:16:02,230 --> 00:16:03,420 probability of going to state two. 255 00:16:03,420 --> 00:16:08,590 Multiply by the payout in that state two. 256 00:16:08,590 --> 00:16:14,280 Sum them up and you get the expression, as I said, the 257 00:16:14,280 --> 00:16:19,280 common thing to choose the strikes is that the contract 258 00:16:19,280 --> 00:16:22,870 has zero value now, so you get your strike. 259 00:16:22,870 --> 00:16:25,760 Well, in particular you could say that if you research the 260 00:16:25,760 --> 00:16:28,400 market well and you know that the stock has equal 261 00:16:28,400 --> 00:16:33,200 probability of going up and down, then actually you expect 262 00:16:33,200 --> 00:16:40,400 your strike to be an average of end values of the stock. 263 00:16:40,400 --> 00:16:44,060 But as we can imagine, following our bookie example, 264 00:16:44,060 --> 00:16:46,270 this is not the right price. 265 00:16:46,270 --> 00:16:50,210 There is actually a definite price which doesn't depend on 266 00:16:50,210 --> 00:16:52,580 condition probability. 267 00:16:52,580 --> 00:16:57,220 Here is the reason why there is a definite price. 268 00:16:57,220 --> 00:17:02,400 Well let's just consider a very simple strategy. 269 00:17:02,400 --> 00:17:05,960 Let's borrow just enough to purchase a stock. 270 00:17:05,960 --> 00:17:10,260 So let's borrow f zero dollars right now and buy the stock 271 00:17:10,260 --> 00:17:12,110 for this money. 272 00:17:12,110 --> 00:17:13,920 And let's enter the forward contract. 273 00:17:13,920 --> 00:17:17,410 Well, by definition forward contract has price zero now, 274 00:17:17,410 --> 00:17:20,440 so we enter the forward contract. 275 00:17:20,440 --> 00:17:23,320 Now, at the time dt when our contract 276 00:17:23,320 --> 00:17:25,530 expires, what happens? 277 00:17:25,530 --> 00:17:29,220 Well, we deliver our stock, which we already have in our 278 00:17:29,220 --> 00:17:32,970 hand for an exchange of k dollars. 279 00:17:32,970 --> 00:17:35,220 That's our forward contract. 280 00:17:35,220 --> 00:17:39,390 On the other hand, we have to repay our loan, and because it 281 00:17:39,390 --> 00:17:40,580 was a loan, it grew. 282 00:17:40,580 --> 00:17:46,520 It grew to s zero times e to the rdt. 283 00:17:46,520 --> 00:17:47,870 Now let's see. 284 00:17:47,870 --> 00:17:52,110 What would happen if k was greater than f 285 00:17:52,110 --> 00:17:54,460 times e to the rdt? 286 00:17:54,460 --> 00:18:00,140 Then we know for sure, we know now for sure, that we would 287 00:18:00,140 --> 00:18:02,460 make money. 288 00:18:02,460 --> 00:18:06,060 There is no uncertainty about it now. 289 00:18:06,060 --> 00:18:10,000 Similarly, if k is less than this value, then we know that 290 00:18:10,000 --> 00:18:12,550 we will lose money. 291 00:18:12,550 --> 00:18:14,670 That's not how the rational market works. 292 00:18:14,670 --> 00:18:18,500 If everybody knew that by setting this price you would 293 00:18:18,500 --> 00:18:21,600 make money, people would do it all day long and 294 00:18:21,600 --> 00:18:23,760 make infinite money. 295 00:18:23,760 --> 00:18:26,720 So there will be no other side of the market. 296 00:18:26,720 --> 00:18:28,280 So the price had to go down. 297 00:18:28,280 --> 00:18:35,460 So the only choice for k, the only market-implied choice, is 298 00:18:35,460 --> 00:18:42,640 that k has to be equal to f times e to the rdt. 299 00:18:42,640 --> 00:18:44,580 As you can see, it doesn't depend on transition 300 00:18:44,580 --> 00:18:46,200 probabilities at all. 301 00:18:46,200 --> 00:18:49,560 That's what market implies us. 302 00:18:49,560 --> 00:18:52,990 That's the price of forward contract, and that actually 303 00:18:52,990 --> 00:18:59,130 explains why when I was watching the forward contract, 304 00:18:59,130 --> 00:19:02,620 current price was just the straight line, it's just 305 00:19:02,620 --> 00:19:06,890 discounted payoff. 306 00:19:06,890 --> 00:19:15,630 The payout is linear, so just the parallel to the payoff. 307 00:19:15,630 --> 00:19:17,690 That's the idea basically. 308 00:19:17,690 --> 00:19:25,620 The idea is to try to find such a portfolio of stock and 309 00:19:25,620 --> 00:19:30,830 the money market account with such a payout , which will 310 00:19:30,830 --> 00:19:35,620 exactly replicate the payoff of our derivative. 311 00:19:35,620 --> 00:19:37,890 If we found such of a portfolio, than we know for 312 00:19:37,890 --> 00:19:39,860 sure that the value of this portfolio, the replicating 313 00:19:39,860 --> 00:19:43,240 portfolio today is equal to the value of the derivative, 314 00:19:43,240 --> 00:19:48,110 because otherwise, you would make or lose money riskless. 315 00:19:48,110 --> 00:19:51,550 That's no-arbitrage condition. 316 00:19:51,550 --> 00:19:56,300 So, can we apply it to our general one step world? 317 00:19:56,300 --> 00:20:01,910 Well, if we have a general payout f, what we want to do, 318 00:20:01,910 --> 00:20:05,350 we want to form a replicating portfolio such that at 319 00:20:05,350 --> 00:20:09,970 expiration time, it will replicate our payouts. 320 00:20:09,970 --> 00:20:15,540 So we want to choose such constants a and b that such 321 00:20:15,540 --> 00:20:18,960 that the combination of stock and money market account in 322 00:20:18,960 --> 00:20:23,570 both states will replicate the payout of our option. 323 00:20:23,570 --> 00:20:28,720 Then if we are able to find such constants a and b, then 324 00:20:28,720 --> 00:20:34,420 we just look at the current price of the contract and it 325 00:20:34,420 --> 00:20:38,930 has to be equal to the current price of our derivative. 326 00:20:38,930 --> 00:20:41,360 Well, but now a particular case, this is easy. 327 00:20:41,360 --> 00:20:45,770 It's just two linear equations with two unknowns, easily 328 00:20:45,770 --> 00:20:52,020 solved, and here is current price of our derivative. 329 00:20:52,020 --> 00:20:53,320 No matter what payout is -- 330 00:20:53,320 --> 00:20:57,210 I mean you just substitute the payout here, and if you know 331 00:20:57,210 --> 00:21:02,980 s1 and s2, and s2 that's it. 332 00:21:02,980 --> 00:21:05,470 A useful way to look at this, just to re-write this 333 00:21:05,470 --> 00:21:12,720 equation, is in this form, and then notice that actually the 334 00:21:12,720 --> 00:21:17,180 current price of our derivative can be viewed as a 335 00:21:17,180 --> 00:21:24,050 discounted expected payout of the derivative, but with very 336 00:21:24,050 --> 00:21:25,350 certain probability. 337 00:21:25,350 --> 00:21:29,240 This probability, it doesn't come from statistical 338 00:21:29,240 --> 00:21:32,870 properties of the stock or from any research, it actually 339 00:21:32,870 --> 00:21:34,720 is defined by the market. 340 00:21:34,720 --> 00:21:37,080 So it's called a risk-neutral probability. 341 00:21:37,080 --> 00:21:44,480 So this probability doesn't depend on the views on the 342 00:21:44,480 --> 00:21:48,660 market by the market participants. 343 00:21:48,660 --> 00:21:53,080 An interesting observation is that actually the value of 344 00:21:53,080 --> 00:21:59,160 this stock, the discounted value of the stock is actually 345 00:21:59,160 --> 00:22:04,190 also is expected value of our outcomes on this risk-neutral 346 00:22:04,190 --> 00:22:06,850 probability. 347 00:22:06,850 --> 00:22:09,450 That's basically general idea. 348 00:22:09,450 --> 00:22:16,220 Now let's move one notch up and try to apply these idea to 349 00:22:16,220 --> 00:22:17,760 continuous case. 350 00:22:17,760 --> 00:22:20,890 Well, if you live in continuous world now, we need 351 00:22:20,890 --> 00:22:27,270 to make some assumptions on the behavior of the stock. 352 00:22:27,270 --> 00:22:31,225 The very common assumption is that the dynamics of the stock 353 00:22:31,225 --> 00:22:32,770 is lognormal. 354 00:22:32,770 --> 00:22:36,060 Lognormal meaning that the logrithm of the stock is 355 00:22:36,060 --> 00:22:38,300 actually normally distributed. 356 00:22:38,300 --> 00:22:43,180 So, here u is some drift, sigma is the volatility of our 357 00:22:43,180 --> 00:22:47,850 stock, and dw is the [INAUDIBLE] process, w's the 358 00:22:47,850 --> 00:22:50,440 [INAUDIBLE] of process such that dw is normally 359 00:22:50,440 --> 00:22:56,320 distributed with mean zero and variance square root dt. 360 00:22:56,320 --> 00:23:00,770 Our approach would be to find the replicating portfolio. 361 00:23:00,770 --> 00:23:05,220 What that mean, it means that we want to find such constants 362 00:23:05,220 --> 00:23:07,020 over time dt. 363 00:23:07,020 --> 00:23:10,900 So we assume that a and b are constant over the next step, 364 00:23:10,900 --> 00:23:15,730 dt, such that the change in our derivative is a linear 365 00:23:15,730 --> 00:23:20,120 combination with this constant of the change of our 366 00:23:20,120 --> 00:23:22,660 underlying security and the change of 367 00:23:22,660 --> 00:23:25,740 money market account. 368 00:23:25,740 --> 00:23:31,800 Now we just need to look more closely at this equation. 369 00:23:31,800 --> 00:23:34,520 First of all, let's concentrate on df. 370 00:23:34,520 --> 00:23:46,070 So, f, our derivative is a function of 371 00:23:46,070 --> 00:23:48,660 stock value you time. 372 00:23:48,660 --> 00:23:53,110 But unfortunately, our stock value is stochastic, so the f 373 00:23:53,110 --> 00:24:00,190 is not that simple, and to write dfo we have to use a 374 00:24:00,190 --> 00:24:03,510 famous -- it's a formula from stochastic calculus, which 375 00:24:03,510 --> 00:24:10,370 actually is analagous of Taylor's formula for 376 00:24:10,370 --> 00:24:12,080 stochastic variables. 377 00:24:12,080 --> 00:24:12,710 Let's see. 378 00:24:12,710 --> 00:24:15,530 If our f will not be stochasticated, it would be 379 00:24:15,530 --> 00:24:19,640 completely deterministic and depend only on dt, then there 380 00:24:19,640 --> 00:24:25,460 would be no term and differential f is just the 381 00:24:25,460 --> 00:24:26,800 standard expression. 382 00:24:26,800 --> 00:24:32,860 On the other hand, if we have dependence on stochastic 383 00:24:32,860 --> 00:24:36,440 variables, then we have to have more terms, 384 00:24:36,440 --> 00:24:38,130 and why this happens. 385 00:24:38,130 --> 00:24:41,480 Well, in very rough words is that because the order of 386 00:24:41,480 --> 00:24:45,510 magnitude of dw is higher than dt's -- 387 00:24:45,510 --> 00:24:47,930 its square root of dt. 388 00:24:47,930 --> 00:24:52,520 So we have to make into account more terms, and 389 00:24:52,520 --> 00:24:55,240 particularly we have to -- 390 00:24:55,240 --> 00:24:59,070 to take in part next order of df squared. 391 00:24:59,070 --> 00:25:02,500 Formally, df square can be written this way, and again, 392 00:25:02,500 --> 00:25:04,900 very rough explanation is as follows. 393 00:25:04,900 --> 00:25:08,460 If we would square this equation there will be three 394 00:25:08,460 --> 00:25:09,490 terms there. 395 00:25:09,490 --> 00:25:13,180 One would come from the square of this term, and this would 396 00:25:13,180 --> 00:25:18,160 be of the order of dt squared, next order of magnitude -- 397 00:25:18,160 --> 00:25:20,240 much smaller than dt. 398 00:25:20,240 --> 00:25:23,060 The second term will be cross-product of dw dt. 399 00:25:23,060 --> 00:25:27,330 What order of magnitude that we are talking about, this is 400 00:25:27,330 --> 00:25:32,990 dt to the power 3/2, again, much smaller than dt. 401 00:25:32,990 --> 00:25:37,090 On the other hand, the third term will be the square of dw, 402 00:25:37,090 --> 00:25:43,500 this is the order of magnitude of dt, so that's what dw is. 403 00:25:43,500 --> 00:25:48,990 And that's what either formula is about. 404 00:25:48,990 --> 00:25:57,730 Now we are basically, now all terms here, and let me stress 405 00:25:57,730 --> 00:26:02,100 out that this term, dv it is not stochastic, it's 406 00:26:02,100 --> 00:26:06,130 completely deterministic because we know that d grows 407 00:26:06,130 --> 00:26:09,350 with the rate r, that's what it is. 408 00:26:09,350 --> 00:26:11,900 So we substitute all those terms into 409 00:26:11,900 --> 00:26:14,510 our replicating equation. 410 00:26:14,510 --> 00:26:16,000 We collect the terms. 411 00:26:16,000 --> 00:26:17,120 We get this equation. 412 00:26:17,120 --> 00:26:20,190 And again, there is the deterministic part, there is 413 00:26:20,190 --> 00:26:20,850 stochastic part. 414 00:26:20,850 --> 00:26:25,530 So the only way for this equation to hold is this term 415 00:26:25,530 --> 00:26:28,300 to be equal to this term, and this term to be equal to this 416 00:26:28,300 --> 00:26:32,170 term, and that's what's written out here. 417 00:26:32,170 --> 00:26:33,580 So again, two equations, these two 418 00:26:33,580 --> 00:26:35,660 unknowns, and here is answer. 419 00:26:35,660 --> 00:26:38,210 420 00:26:38,210 --> 00:26:47,800 Finally, let's take af to another part. 421 00:26:47,800 --> 00:26:51,300 Notice that this part of our equation is completely 422 00:26:51,300 --> 00:26:53,880 deterministic. 423 00:26:53,880 --> 00:26:55,810 So we know how it will grow. 424 00:26:55,810 --> 00:27:02,790 So basically, d of f minus as, which is d times dv, is r 425 00:27:02,790 --> 00:27:06,190 times d times dc. 426 00:27:06,190 --> 00:27:09,710 We know all other terms, we substitute them here, take 427 00:27:09,710 --> 00:27:13,050 something to the left hand side and 428 00:27:13,050 --> 00:27:14,220 we have this equation. 429 00:27:14,220 --> 00:27:20,730 So this is partial differential equation for our 430 00:27:20,730 --> 00:27:27,550 derivative f, as a function of f and t, of second order, and 431 00:27:27,550 --> 00:27:30,310 this equation is the famous Black-Scholes equation. 432 00:27:30,310 --> 00:27:35,390 It was derived by Fisher Black and Myron Scholes in their 433 00:27:35,390 --> 00:27:38,940 famous paper published in 1973. 434 00:27:38,940 --> 00:27:41,880 Myron Scholes and Robert Merton actually received Nobel 435 00:27:41,880 --> 00:27:47,560 Prize for deriving and solving this equation in '97. 436 00:27:47,560 --> 00:27:51,250 Black was already dead by the time. 437 00:27:51,250 --> 00:27:55,320 This is really the cornerstone of math finance. 438 00:27:55,320 --> 00:27:59,290 439 00:27:59,290 --> 00:28:04,560 The cornerstone is because using the replicating 440 00:28:04,560 --> 00:28:09,370 portfolio, using this reasoning, we were able to 441 00:28:09,370 --> 00:28:14,040 find an exact equation for our derivative. 442 00:28:14,040 --> 00:28:16,980 So a few would remarks on Black-Scholes. 443 00:28:16,980 --> 00:28:21,680 So first of all, we make some assumptions on the dynamic of 444 00:28:21,680 --> 00:28:26,120 the stock, but we never made any assumptions on our 445 00:28:26,120 --> 00:28:27,150 derivative. 446 00:28:27,150 --> 00:28:31,190 Which means that any derivative has to satisfy this 447 00:28:31,190 --> 00:28:34,780 equation, and that's very strong result. 448 00:28:34,780 --> 00:28:37,140 So if you assume that our stock was normal, which is not 449 00:28:37,140 --> 00:28:40,620 a bad assumption and agrees quite well with the market, 450 00:28:40,620 --> 00:28:44,470 then we basically in principle can price any derivative. 451 00:28:44,470 --> 00:28:48,400 We know the equation for any derivative. 452 00:28:48,400 --> 00:28:52,180 The other thing is that our Black-Scholes equation doesn't 453 00:28:52,180 --> 00:28:53,740 depend on the actual [UNINTELLIGIBLE] 454 00:28:53,740 --> 00:28:55,530 [? mu ?] 455 00:28:55,530 --> 00:28:57,180 in the dynamics of our stock. 456 00:28:57,180 --> 00:29:06,320 So again, it is the manifest of risk-neutral dynamic. 457 00:29:06,320 --> 00:29:12,150 Not only we wrote down the equation for our derivative, 458 00:29:12,150 --> 00:29:15,030 we also found a replicating portfolio. 459 00:29:15,030 --> 00:29:18,670 So in other words, we found a hedge and strategy, meaning 460 00:29:18,670 --> 00:29:24,350 that at any given time we can form this portfolio 461 00:29:24,350 --> 00:29:28,360 with rates a and b. 462 00:29:28,360 --> 00:29:33,460 If we hold both the derivative and both replicating portfolio 463 00:29:33,460 --> 00:29:38,490 all together, this is zero sum gain. 464 00:29:38,490 --> 00:29:42,600 We know that no matter where stock moves, we will not move 465 00:29:42,600 --> 00:29:44,710 money or gain money. 466 00:29:44,710 --> 00:29:51,400 So if we just charge bid offer on the derivative, if we 467 00:29:51,400 --> 00:29:54,500 charge a few on the contract, we can hedge ourself 468 00:29:54,500 --> 00:29:57,740 perfectly, buy the contract or sell the contract, hedge 469 00:29:57,740 --> 00:30:00,510 perfectly ourself and just make money on 470 00:30:00,510 --> 00:30:01,760 the fee, that's it. 471 00:30:01,760 --> 00:30:06,180 472 00:30:06,180 --> 00:30:11,330 Finally, more mathematical remark is that actually after 473 00:30:11,330 --> 00:30:13,960 a few manipulations, a few change of variables, the 474 00:30:13,960 --> 00:30:17,870 Black-Scholes equation comes out to be just a heat 475 00:30:17,870 --> 00:30:22,190 question, which you already saw in this class. 476 00:30:22,190 --> 00:30:23,650 This is very good news. 477 00:30:23,650 --> 00:30:25,000 Why is this good news? 478 00:30:25,000 --> 00:30:27,250 Well, because heat equation is very well studied. 479 00:30:27,250 --> 00:30:32,000 So the solutions are well-known, and numerical 480 00:30:32,000 --> 00:30:34,640 methods, the ways to solve it, and particular the numerical 481 00:30:34,640 --> 00:30:36,760 ways to solve it are well-known. 482 00:30:36,760 --> 00:30:38,220 So we are in business. 483 00:30:38,220 --> 00:30:45,350 But as any partial differential equation, the 484 00:30:45,350 --> 00:30:49,680 equation itself doesn't make much sense because to find a 485 00:30:49,680 --> 00:30:51,720 particular solution we need 486 00:30:51,720 --> 00:30:55,610 boundary and initial condition. 487 00:30:55,610 --> 00:30:58,520 488 00:30:58,520 --> 00:31:01,940 And although any derivative which defines Black-Scholes 489 00:31:01,940 --> 00:31:09,230 equation, the final and boundary condition will vary 490 00:31:09,230 --> 00:31:10,720 from contract to contract. 491 00:31:10,720 --> 00:31:15,510 Here are a few examples of the final and boundary conditions. 492 00:31:15,510 --> 00:31:19,310 Here, an interesting remark that usually we would talk 493 00:31:19,310 --> 00:31:24,250 about initial condition, here we are talking about final 494 00:31:24,250 --> 00:31:26,730 condition if time goes in reverse -- we know the state 495 00:31:26,730 --> 00:31:30,150 of the world at the end, at expiration, not today. 496 00:31:30,150 --> 00:31:35,505 So, here are final and boundary conditions for call 497 00:31:35,505 --> 00:31:40,390 and put, and let's look a little bit at the pictures for 498 00:31:40,390 --> 00:31:44,250 our call and put to see where they come from. 499 00:31:44,250 --> 00:31:47,930 So for example, for calls, well, this is our final 500 00:31:47,930 --> 00:31:51,060 condition, right, this is defined by the payout. 501 00:31:51,060 --> 00:31:55,000 Under the boundary condition, well, what happens, we put 502 00:31:55,000 --> 00:31:57,530 them at zero at an infinity, which shows 503 00:31:57,530 --> 00:32:00,980 [INAUDIBLE PHRASE], and why, well, because if stock hits 504 00:32:00,980 --> 00:32:03,340 zero then it stays in zero. 505 00:32:03,340 --> 00:32:06,330 That's what our dynamics show. 506 00:32:06,330 --> 00:32:10,750 So the value of power of contract at maturity will 507 00:32:10,750 --> 00:32:11,400 become zero. 508 00:32:11,400 --> 00:32:12,240 [UNINTELLIGIBLE PHRASE] 509 00:32:12,240 --> 00:32:15,740 the stock grows, grows to inifinity, a good assumption 510 00:32:15,740 --> 00:32:18,410 to make is that actually it becomes similar to stock 511 00:32:18,410 --> 00:32:22,070 itself, so it would just become parallel to the stock, 512 00:32:22,070 --> 00:32:27,150 and that's the conditions between both here. 513 00:32:27,150 --> 00:32:30,270 514 00:32:30,270 --> 00:32:37,060 Similar, for the put you can derive these conditions, and 515 00:32:37,060 --> 00:32:41,740 again, that's because it is a heat equation. 516 00:32:41,740 --> 00:32:44,910 It turns out that for a simple derivative such that the calls 517 00:32:44,910 --> 00:32:47,050 and puts, it is possible to find an 518 00:32:47,050 --> 00:32:50,070 exact analytic solution. 519 00:32:50,070 --> 00:32:54,630 Here are exactimated set of solutions for a call, put and 520 00:32:54,630 --> 00:32:57,090 the digital contracts. 521 00:32:57,090 --> 00:33:03,640 Well, not surprising again, I mean they're all connected to 522 00:33:03,640 --> 00:33:07,100 the error function, so to the normal distribution, 523 00:33:07,100 --> 00:33:12,400 basically, as the solutions of heat equation ought to be. 524 00:33:12,400 --> 00:33:14,580 Why do they look exactly the same? 525 00:33:14,580 --> 00:33:19,410 If we have five minutes at the end, we'll probably shed some 526 00:33:19,410 --> 00:33:22,750 light on the specific form of equations. 527 00:33:22,750 --> 00:33:28,560 But just trust that we can see that it's discounted, and what 528 00:33:28,560 --> 00:33:31,580 I'm saying is expected value of our 529 00:33:31,580 --> 00:33:33,520 payout on the [? risk ?] 530 00:33:33,520 --> 00:33:36,810 [UNINTELLIGIBLE] for measure. 531 00:33:36,810 --> 00:33:42,440 Here is an example -- it's of a particular call option on 532 00:33:42,440 --> 00:33:44,160 the same IBM stock. 533 00:33:44,160 --> 00:33:48,650 So I chose the short-dated contract, just avoid the 534 00:33:48,650 --> 00:33:50,300 dividend payment. 535 00:33:50,300 --> 00:33:53,820 So it's a contract expiring on March 18. 536 00:33:53,820 --> 00:33:56,750 So there is some based expiration. 537 00:33:56,750 --> 00:34:02,530 The stock, as we saw, the expirations of stock, as we 538 00:34:02,530 --> 00:34:06,750 saw, is what's trading at 81.14. 539 00:34:06,750 --> 00:34:12,820 The volatility is somewhere around 14%, estimated either 540 00:34:12,820 --> 00:34:17,110 from other options or from historically. 541 00:34:17,110 --> 00:34:20,210 Here is the price of our contract. 542 00:34:20,210 --> 00:34:23,750 I also have a simple Black-Scholes calculator here, 543 00:34:23,750 --> 00:34:28,800 and let's see if we can match this price. 544 00:34:28,800 --> 00:34:35,600 So let's see, I believe the volatility 545 00:34:35,600 --> 00:34:42,240 was 13%, right, 13.47. 546 00:34:42,240 --> 00:34:45,220 The interest rate, it's already here. 547 00:34:45,220 --> 00:34:48,750 As we all know the Fed just bumped the interest rate, so 548 00:34:48,750 --> 00:34:51,980 at 4.75% right now. 549 00:34:51,980 --> 00:34:57,700 The striker of our option was 80 times expiration was 550 00:34:57,700 --> 00:35:00,480 actually 10 days, and this should be measured at a 551 00:35:00,480 --> 00:35:02,390 fraction of year. 552 00:35:02,390 --> 00:35:08,440 So we divide 10 by 365. 553 00:35:08,440 --> 00:35:15,350 This stock was trading at 81.14, if I'm not mistaken. 554 00:35:15,350 --> 00:35:17,690 Here is the price of our call options 555 00:35:17,690 --> 00:35:21,210 contract, which is 150. 556 00:35:21,210 --> 00:35:26,630 Well, it's within the offer. 557 00:35:26,630 --> 00:35:31,540 So maybe our volatility's slightly off and if increase 558 00:35:31,540 --> 00:35:42,330 at, let's say, 2.14% it will go slightly up. 559 00:35:42,330 --> 00:35:44,800 152. 560 00:35:44,800 --> 00:35:48,580 Well, in general, let's play a little bit with it. 561 00:35:48,580 --> 00:35:51,690 Well, it is very short-dated option, so the value of our 562 00:35:51,690 --> 00:35:55,990 option is very close to the payout. 563 00:35:55,990 --> 00:35:59,630 So if we increase the time to maturity, let's make it three 564 00:35:59,630 --> 00:36:06,150 years just to see where, so now what if our option is -- 565 00:36:06,150 --> 00:36:08,980 well, that's what it is. 566 00:36:08,980 --> 00:36:15,040 If increase volatility, sure enough, let's make it 30%. 567 00:36:15,040 --> 00:36:15,970 So what do we expect? 568 00:36:15,970 --> 00:36:18,960 We expect if volatility higher, then certain things 569 00:36:18,960 --> 00:36:21,500 higher, so the value of our contract should go 570 00:36:21,500 --> 00:36:25,300 up, and it sure does. 571 00:36:25,300 --> 00:36:29,120 572 00:36:29,120 --> 00:36:31,520 So basically that's how Black-Scholes works. 573 00:36:31,520 --> 00:36:39,690 574 00:36:39,690 --> 00:36:39,950 [? Plane ?] 575 00:36:39,950 --> 00:36:45,150 [? Hills ?] does contract trade on the market, but 576 00:36:45,150 --> 00:36:48,610 unfortunately not all of these contracts are so simple as 577 00:36:48,610 --> 00:36:50,600 calls and puts. 578 00:36:50,600 --> 00:36:57,280 First of all, there are many more complicated products with 579 00:36:57,280 --> 00:37:03,760 more difficult payout, which will constitute different and 580 00:37:03,760 --> 00:37:07,790 [UNINTELLIGIBLE] will discontinue final conditions 581 00:37:07,790 --> 00:37:10,930 on our Black-Scholes equation. 582 00:37:10,930 --> 00:37:14,590 Moreover, we made an assumption that the volatility 583 00:37:14,590 --> 00:37:16,730 is constant with time, and interest rate is 584 00:37:16,730 --> 00:37:18,210 constant with time. 585 00:37:18,210 --> 00:37:21,860 It is certainly not true for the real world. 586 00:37:21,860 --> 00:37:25,580 Volatility probably should be time dependent, and this would 587 00:37:25,580 --> 00:37:27,390 make the coefficients in our Black-Scholes 588 00:37:27,390 --> 00:37:30,090 equation time dependent. 589 00:37:30,090 --> 00:37:33,250 Unfortunately, these cannot be solve analytically. 590 00:37:33,250 --> 00:37:37,310 So in most of the cases in practice, we will have to use 591 00:37:37,310 --> 00:37:42,530 some kind of numerical solution. 592 00:37:42,530 --> 00:37:46,430 Finding different methods is the typical approach for the 593 00:37:46,430 --> 00:37:47,200 heat equation. 594 00:37:47,200 --> 00:37:49,450 As you know, both [UNINTELLIGIBLE PHRASE] 595 00:37:49,450 --> 00:37:57,910 schemes, and you will discuss some of those in 18.086. 596 00:37:57,910 --> 00:37:58,630 Tree methods. 597 00:37:58,630 --> 00:38:05,090 Tree methods meaning that we go back to our one step tree, 598 00:38:05,090 --> 00:38:08,930 and basically assume that our times expiration is many 599 00:38:08,930 --> 00:38:11,170 [UNINTELLIGIBLE] time steps away and will grow the tree 600 00:38:11,170 --> 00:38:14,590 further, so from this node we have some more nodes, and so 601 00:38:14,590 --> 00:38:15,550 and so forth. 602 00:38:15,550 --> 00:38:20,240 That would imply the final condition at the end, and 603 00:38:20,240 --> 00:38:23,640 discount back using risk-neutral probabilities, 604 00:38:23,640 --> 00:38:25,170 and get the price down. 605 00:38:25,170 --> 00:38:27,240 So those are called tree methods. 606 00:38:27,240 --> 00:38:29,810 One can show that actually those tree methods are 607 00:38:29,810 --> 00:38:34,850 equivalent to find a difference -- expect to find a 608 00:38:34,850 --> 00:38:36,100 difference scheme. 609 00:38:36,100 --> 00:38:38,110 Those are very popular. 610 00:38:38,110 --> 00:38:42,625 But again, in tree methods, what is very important is to 611 00:38:42,625 --> 00:38:44,830 set the probability from your tree the transition 612 00:38:44,830 --> 00:38:46,120 probability was the right one. 613 00:38:46,120 --> 00:38:48,960 Since the right one risk-neutral probability. 614 00:38:48,960 --> 00:38:53,170 Probabilities implied by the market, actually. 615 00:38:53,170 --> 00:38:58,280 Another important numerical method is Monte Carlo 616 00:38:58,280 --> 00:39:02,910 simulation where you would simulate many different 617 00:39:02,910 --> 00:39:04,340 scenarios of the development of your 618 00:39:04,340 --> 00:39:06,290 stock after the maturity. 619 00:39:06,290 --> 00:39:13,010 Then basically using this path you will find the expected 620 00:39:13,010 --> 00:39:15,990 value of your payout. 621 00:39:15,990 --> 00:39:22,310 But again, in order for this expected value to be the same 622 00:39:22,310 --> 00:39:26,730 as the risk-neutral value , as the arbitrage-free value, you 623 00:39:26,730 --> 00:39:30,120 have to develop your Monte Carlo simulations is 624 00:39:30,120 --> 00:39:31,800 risk-neutral probability. 625 00:39:31,800 --> 00:39:37,360 So, risk-neutral volition is extremely important. 626 00:39:37,360 --> 00:39:40,780 627 00:39:40,780 --> 00:39:45,810 Here is actually the general risk-neutral statement, which 628 00:39:45,810 --> 00:39:51,920 one can prove is that actually the value of any derivative is 629 00:39:51,920 --> 00:39:56,380 just discounted expected value of the payout of this 630 00:39:56,380 --> 00:40:00,310 derivative at maturity, but you have to take this 631 00:40:00,310 --> 00:40:02,140 expectation and divide measure. 632 00:40:02,140 --> 00:40:04,710 Using the right measure, meaning that you have to set 633 00:40:04,710 --> 00:40:06,990 correctly the transition probability -- 634 00:40:06,990 --> 00:40:09,330 you have to make them market-neutral. 635 00:40:09,330 --> 00:40:12,710 636 00:40:12,710 --> 00:40:19,950 Under this measure, actually the dynamics of our stocks 637 00:40:19,950 --> 00:40:23,400 looks slightly different, and as you can see, our drift 638 00:40:23,400 --> 00:40:25,240 becomes the interest rate. 639 00:40:25,240 --> 00:40:26,860 So either [UNINTELLIGIBLE PHRASE] 640 00:40:26,860 --> 00:40:29,510 you measure, everything grows with our 641 00:40:29,510 --> 00:40:32,430 risk-free interest rate. 642 00:40:32,430 --> 00:40:39,660 Just to shed a little bit of light on how we go at the 643 00:40:39,660 --> 00:40:43,880 solutions for calls and puts, Black Shoals solutions for 644 00:40:43,880 --> 00:40:48,160 call and put, well this is the distribution of our stock, the 645 00:40:48,160 --> 00:40:52,250 normal distribution of our stock at time t, and if we 646 00:40:52,250 --> 00:40:58,230 take this distribution and integrate our payout of our 647 00:40:58,230 --> 00:41:00,270 co-option against this distribution -- in other 648 00:41:00,270 --> 00:41:09,680 words, find the expected value of payout of our full option 649 00:41:09,680 --> 00:41:18,000 under the risk-neutral measure, then sure enough you 650 00:41:18,000 --> 00:41:21,200 will get [UNINTELLIGIBLE PHRASE]. 651 00:41:21,200 --> 00:41:24,000 This illustrates the best because what is digital -- 652 00:41:24,000 --> 00:41:27,370 digital is just the probability to end up at both 653 00:41:27,370 --> 00:41:29,590 a strike at time t, right? 654 00:41:29,590 --> 00:41:36,490 So if you integrate this lognormal pdf from the strike 655 00:41:36,490 --> 00:41:42,310 k to the infinite, that will be your answer. 656 00:41:42,310 --> 00:41:45,590 This is a good exercise in integration to make sure that 657 00:41:45,590 --> 00:41:47,760 it's correct. 658 00:41:47,760 --> 00:41:49,270 So let's see. 659 00:41:49,270 --> 00:41:52,170 To conclude, what we've seen. 660 00:41:52,170 --> 00:42:01,840 So, we have seen that more than derivatives business 661 00:42:01,840 --> 00:42:06,220 makes use of quite advanced mathematics, and what kinds of 662 00:42:06,220 --> 00:42:08,050 mathematics is used there. 663 00:42:08,050 --> 00:42:12,630 Well, partial differential equations are used heavily. 664 00:42:12,630 --> 00:42:15,730 Numerical methods for the solution of this partial 665 00:42:15,730 --> 00:42:20,470 differential equations are naturally used. 666 00:42:20,470 --> 00:42:24,180 In order to get the integrations, we actually need 667 00:42:24,180 --> 00:42:28,190 to operate in some stochastic calculus, meaning that we need 668 00:42:28,190 --> 00:42:31,550 to know how to deal with either calculus, either 669 00:42:31,550 --> 00:42:36,580 formula, Gaussian theorem, and so on and so forth. 670 00:42:36,580 --> 00:42:41,990 The other thing is to be able to build simulations to solve 671 00:42:41,990 --> 00:42:45,380 the heat equation and all other equations 672 00:42:45,380 --> 00:42:47,780 that you might encounter. 673 00:42:47,780 --> 00:42:52,120 The topic which we didn't touch upon is statistics 674 00:42:52,120 --> 00:42:56,010 because, of course, very advanced statistics is used 675 00:42:56,010 --> 00:43:01,330 for many, many things for analyzing historical data, 676 00:43:01,330 --> 00:43:06,250 which can be quite beautiful for trading strategies and 677 00:43:06,250 --> 00:43:07,660 many others. 678 00:43:07,660 --> 00:43:13,630 Besides these five topics, there is much, much more to 679 00:43:13,630 --> 00:43:17,760 mathematical finance, which makes it a very, very exciting 680 00:43:17,760 --> 00:43:19,760 field to work in. 681 00:43:19,760 --> 00:43:22,240 That's what I wanted to talk about. 682 00:43:22,240 --> 00:43:23,750 Thank you very much for your attention. 683 00:43:23,750 --> 00:43:30,890 684 00:43:30,890 --> 00:43:36,010 PROFESSOR: Maybe I'll ask a firm question about boundary 685 00:43:36,010 --> 00:43:40,060 conditions, because you had said that those are different 686 00:43:40,060 --> 00:43:46,000 for different contracts and how do you deal with them in 687 00:43:46,000 --> 00:43:52,140 the finite differences or the tree model or whatever. 688 00:43:52,140 --> 00:43:54,750 What would be a one typical one? 689 00:43:54,750 --> 00:43:57,560 GUEST SPEAKER: Well, typical one -- two very typical ones. 690 00:43:57,560 --> 00:44:05,640 So those you basically make a grid of your problem, in 691 00:44:05,640 --> 00:44:09,915 particular you build a tree, which is actually a grid of 692 00:44:09,915 --> 00:44:11,650 all possible outcomes. 693 00:44:11,650 --> 00:44:16,820 You set them up as the ends, so your tree grows. 694 00:44:16,820 --> 00:44:28,490 695 00:44:28,490 --> 00:44:35,360 So you set your boundaries here at the end, and well you 696 00:44:35,360 --> 00:44:41,070 set probably some initial -- 697 00:44:41,070 --> 00:44:44,560 698 00:44:44,560 --> 00:44:46,840 this is final condition, so you set some boundary 699 00:44:46,840 --> 00:44:48,690 conditions here. 700 00:44:48,690 --> 00:44:50,890 So this is your time t. 701 00:44:50,890 --> 00:44:57,230 This is t zero times t -- this is zero, this is 1, this 702 00:44:57,230 --> 00:45:00,830 is 2, this is t. 703 00:45:00,830 --> 00:45:08,970 So you set your payout here, so it will be maximum of s 704 00:45:08,970 --> 00:45:11,340 minus k and zero. 705 00:45:11,340 --> 00:45:14,940 PROFESSOR: How many time steps might you take in this? 706 00:45:14,940 --> 00:45:20,315 GUEST SPEAKER: Well, you would do like daily for three months 707 00:45:20,315 --> 00:45:22,690 -- you do three month options. 708 00:45:22,690 --> 00:45:23,600 PROFESSOR: Maybe 100 steps. 709 00:45:23,600 --> 00:45:24,960 GUEST SPEAKER: Yeah, something like that. 710 00:45:24,960 --> 00:45:28,470 Well, if it's two year option, that you probably would do it 711 00:45:28,470 --> 00:45:30,820 weekly or something like that. 712 00:45:30,820 --> 00:45:33,220 PROFESSOR: You don't get into large, what would be 713 00:45:33,220 --> 00:45:35,490 scientifically, large-scale calculating. 714 00:45:35,490 --> 00:45:37,490 PROFESSOR: No, in finance you usually 715 00:45:37,490 --> 00:45:39,040 don't keep this problem--. 716 00:45:39,040 --> 00:45:42,350 PROFESSOR: In finite differences, do you use like 717 00:45:42,350 --> 00:45:45,970 higher order as opposed, where you had second derivatives, 718 00:45:45,970 --> 00:45:49,250 would you always use second differences or 719 00:45:49,250 --> 00:45:51,590 second order accuracy? 720 00:45:51,590 --> 00:45:52,740 GUEST SPEAKER: In general, yes. 721 00:45:52,740 --> 00:45:55,790 In general, second order access. 722 00:45:55,790 --> 00:45:58,210 In general you don't go higher. 723 00:45:58,210 --> 00:46:01,920 I mean the precision -- well, it's within tenths, right. 724 00:46:01,920 --> 00:46:06,010 So you can not do better than that. 725 00:46:06,010 --> 00:46:11,390 So it depends -- well, it depends what kind of amount 726 00:46:11,390 --> 00:46:12,830 you are dealing with. 727 00:46:12,830 --> 00:46:16,880 If you're actually selling and buying units of stock, you 728 00:46:16,880 --> 00:46:21,080 might consider something more precise. 729 00:46:21,080 --> 00:46:25,220 But it's very problem-defined. 730 00:46:25,220 --> 00:46:30,280 731 00:46:30,280 --> 00:46:33,370 So that's how we deal with it. 732 00:46:33,370 --> 00:46:35,770 PROFESSOR: Any questions? 733 00:46:35,770 --> 00:46:38,440 We can put the mic on if you have a question. 734 00:46:38,440 --> 00:46:45,080 STUDENT: [UNINTELLIGIBLE PHRASE]. 735 00:46:45,080 --> 00:46:55,800 736 00:46:55,800 --> 00:46:58,140 GUEST SPEAKER: Well, it is market proc-- 737 00:46:58,140 --> 00:46:58,500 Yes. 738 00:46:58,500 --> 00:47:01,480 I mean this is just a numerical solution. 739 00:47:01,480 --> 00:47:02,910 So yeah, it is market process. 740 00:47:02,910 --> 00:47:06,305 and basically all stochastic calculus is about market 741 00:47:06,305 --> 00:47:07,590 process, continuous market process. 742 00:47:07,590 --> 00:47:15,030 743 00:47:15,030 --> 00:47:18,290 PROFESSOR: Is the mathematics that you get involved with 744 00:47:18,290 --> 00:47:25,970 pretty well set now or is there a need for more 745 00:47:25,970 --> 00:47:28,920 mathematics, if I can ask the question that way? 746 00:47:28,920 --> 00:47:31,460 PROFESSOR: Yeah, well in this field it is 747 00:47:31,460 --> 00:47:33,610 probably quite well set. 748 00:47:33,610 --> 00:47:38,340 But if you get into more complicated fields, especially 749 00:47:38,340 --> 00:47:45,080 into credit modeling, the model for the credits of 750 00:47:45,080 --> 00:47:49,810 certain companies, then mathematics is not quite set, 751 00:47:49,810 --> 00:47:53,730 because there you start talking about jump processes 752 00:47:53,730 --> 00:47:54,360 and [UNINTELLIGIBLE] 753 00:47:54,360 --> 00:47:58,860 Weiner processes, not just lognormal processes. 754 00:47:58,860 --> 00:48:03,070 This stochastic differential equations become very hard, 755 00:48:03,070 --> 00:48:06,150 but may be analytically practical. 756 00:48:06,150 --> 00:48:09,330 So from this point of view there is -- 757 00:48:09,330 --> 00:48:13,815 but it's not a fundamental mathematics, it's not that you 758 00:48:13,815 --> 00:48:20,140 are opening a new field, but definitely trying to solve a 759 00:48:20,140 --> 00:48:22,620 stochastic differential equation, which usually boils 760 00:48:22,620 --> 00:48:25,170 down to solving a partial differential equation 761 00:48:25,170 --> 00:48:26,780 analytically. 762 00:48:26,780 --> 00:48:30,465 Can be pretty hard in mathematical problem, view as 763 00:48:30,465 --> 00:48:32,280 a mathematical problem. 764 00:48:32,280 --> 00:48:35,030 PROFESSOR: So you showed the example of 765 00:48:35,030 --> 00:48:37,700 Black-Scholes solver. 766 00:48:37,700 --> 00:48:40,210 Everybody has that available all the time? 767 00:48:40,210 --> 00:48:40,830 GUEST SPEAKER: Oh yeah. 768 00:48:40,830 --> 00:48:45,790 On Chicago trading floor, the traders have calculators where 769 00:48:45,790 --> 00:48:49,650 they just press a button and it's just hard-wired there. 770 00:48:49,650 --> 00:48:51,270 PROFESSOR: And they're printing out error functions, 771 00:48:51,270 --> 00:48:52,230 basically -- 772 00:48:52,230 --> 00:48:55,060 a combination of error function, yeah. 773 00:48:55,060 --> 00:49:01,110 GUEST SPEAKER: Well, sure enough, nobody uses just -- 774 00:49:01,110 --> 00:49:03,620 I mean this was their approximate example and that's 775 00:49:03,620 --> 00:49:08,120 why I chose such short-dated stock that before it pays any 776 00:49:08,120 --> 00:49:12,920 dividends, and where we cannot the volatility is constant, 777 00:49:12,920 --> 00:49:15,400 and so on and so forth, match the prices. 778 00:49:15,400 --> 00:49:28,430 Otherwise, the prices wouldn't match. 779 00:49:28,430 --> 00:49:29,930 PROFESSOR: Thank you. 780 00:49:29,930 --> 00:49:33,139